IEEK Transactions on Smart Processing and Computing, vol. 2, no. 1, February 2013 20

IEEK Transactions on Smart Processing and Computing

Image Noise Reduction in Discrete Cosine Transform domain

Hyosun Joo, Junhee Park, Jeongtae Kim, and Byung-Uk Lee

Department of Electronics Engineering, Ewha Womans University / 11-1 Daehyun Dong, Seoul 120-750, Korea julyhs@naver.com, corelpark@gmail.com, jtkim@ewha.ac.kr, bulee@ewha.ac.kr * Corresponding Author: Byung-Uk Lee

Received January 9, 2013; Revised January 16, 2013; Accepted February 3, 2013; Published February 28, 2013

Abstract: Image noise reduction in the frequency domain by thresholding is simple, but quite effective. Wavelet domain thresholding has been an active area of research but relatively little work has been published on DCT domain denoising. A novel method for determining the hard threshold for the DCT domain denoising is proposed. The low amplitude DCT coefficients are discarded until the cumulative sum of the discarded signal energy is comparable to that of noise in each DCT block. Cycle spinning is also applied to reduce block artifacts. The proposed method is quite effective and simple enough to be used in portable devices.

Keywords: DCT, Hard thresholding, Denoising 1. Introduction

High resolution imaging is accomplished by decreasing the pixel size of an image sensor, which introduces several issues that need to be addressed. One important issue affecting the image quality is noise. With the reduced sensor area, each pixel accumulates fewer photons, making the processing and transfer of the image signal susceptible to noise.

Considerable research on noise reduction has been conducted [1, 2]. A noise reduction technique was investigated in the discrete cosine transform (DCT) domain because the DCT is employed in image and video compression standards, such as JPEG, MPEG, H.261, H.263, and H.264. DCT is quite effective in energy compaction; a few low frequency components retain a major portion of the signal energy, leaving most of the high frequency coefficients at zero. Because the noise power spectrum is distributed evenly at low and high frequencies, noise can be reduced by applying thresholding techniques. As a proper threshold is essential for the performance, methods for selecting wavelet thresholding and threshold value for noise reduction have been studied extensively [5, 6]. Although the techniques to determine the optimal threshold in the wavelet domain are abundant

[5, 7], there are only a few methods for determining the threshold in the DCT domain. Noise reduction in the DCT domain is attractive because DCT is normally applied to image or video compression after image acquisition. In this study, a simple and effective block adaptive DCT threshold decision algorithm is proposed.

This paper is organized as follows. Section 2 introduces existing denoising methods. Section 3 outlines the proposed denoising method. Section 4 presents the experimental results and discussions, and the conclusions are reported in Section 5.

2. Thresholding techniques for denoising

The existing threshold selection methods in the wavelet and DCT domain are introduced

2.1 Wavelet domain threshold determination

One classical method to determine the noise threshold in the wavelet domain is Universal shrink, or Visu shrink [5, 6], which is based on the hard threshold defined as follows:

This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2011-0010378) and SK Hynix.

2 lnUniv n pNλ σ= (1)

IEEK Transactions on Smart Processing and Computing, vol. 2, no. 1, February 2013 21

where nσ is the noise standard deviation, and is the length of the signal or the number of pixels.

pN

Another effective method is based on the Bayesian rule [7, 11], which models the signal distribution as a generalized Gaussian distribution (GGD) and then estimates the soft threshold in the wavelet domain, which results in the minimum mean square error. The threshold is given as follows:

2 ˆ/Bayes n xλ σ σ= , (2)

where ˆ xσ is the variance of the original signal.

2.2 Shape-Adaptive DCT (SADCT) hard thresholding

Another approach is to exploit a shape-adaptive transformation on the neighborhoods whose shapes are adaptive to the salient image features and contain mostly homogeneous signals [4]. The shape-adaptive transformation can achieve a very sparse representation of the true signal in the adaptive neighborhoods.

Shape-adaptive DCT is applied to the locally adaptive window to reconstruct the observed noisy images. The adaptive window is chosen by an anisotropic local polynomial approximation intersection confidence interval (LPA-ICI). The adaptive window can be obtained locally for each pixel, manipulating eight directional length-varying LPA kernel estimates. In other words, every pixel has eight directional kernels. Denoising is accomplished by employing sequentially SADCT, hard thresholding, and inverse SADCT in the window. This method is free from blocking artifacts due to the feature adaptive DCT windows [4].

2.3 Generalized Gaussian Markov Random Field (GGMRF) – DCT method

The GGMRF method uses both thresholding in the DCT domain and generalized Gaussian Markov Random Field prior to an image restoration and denoising [3]. The threshold level is proportional to the image noise and a regularization function is used to preserve the edges during image restoration. Cycle spinning is also applied to reduce the blocking artifacts. The algorithm requires at least 400 iterations to obtain a numerical solution. The performance is comparable to the state of the art algorithms [13].

3. Determination of Threshold Using Signal Energy

Thresholding operation on the DCT coefficients can be applied to image for denoising. Discontinuity artifacts along the block boundary are unavoidable because DCT is based on block processing. Many studies have examined reducing the block artifacts [14-16]. On the other hand, cycle spinning [17] was adopted because it can ameliorate

those block artifacts. Cycle spinning applies a circular shift to the image, which results in a shift in the block boundaries, and then applies denoising. The average of the shifted, denoised and inverse shifted images were taken, which reduces the block artifacts and noise further. Eq. (10) in Section 3 shows the precise details.

The observed image block has an additive zero mean Gaussian noise 2~ (0, )nn N σ as follows:

y x n= + , (3)

where is an observed image, and y x denotes an original image. Assume that the noise is statistically independent from the signal. The observed signal energy can be approximated as follows:

(4)

2 2

2 2

2 2

E[ ] = E[ ] 2E[ ] E[ ] = E[ ] E[ ] = E[ ] . n

y x xn nx nx σ

+ +

+

+

2

The purpose of this study was to determine a threshold

level that separates the large amplitude signals from small amplitude noise. The DCT coefficients of an image block were sorted and observed at a level where the sum of the energy of the thresholded energy is equal to the noise energy. The threshold level can be found using the estimated noise variance from an observed image [7-10].

The operator H in the following equation denotes the DCT transformation operator. Therefore, yH represents the transformed coefficients. The DCT coefficients are sorted in ascending order of the absolute value as follows:

(5) 1 2

1 2

= {|c | ,|c | ,...,|c | }, where |c | | c | ... | c |,

N N

N N

y ×

×≤ ≤ ≤

H

, (6) 2

1

( ) c m

vv

F m=

= ∑

where indicates the DCT coefficient and cv ( )F m is the energy of the thresholded signal when the threshold level λ is set to as follows: cm

2 2( ) [ ] ,

| c |.n

m

F m E n σλ

= ≈

=, (7)

On the other hand, noise can be reduced further by

scaling the threshold level, i.e.

. (8) 2( ) nF m κσ= The optimal κ was calculated from simulations with

16 images. Fig. 1 shows that the optimal κ value is approximately 1.1 when the noise standard deviation is 20 (Fig. 1(a)) and 10 (Fig. 1(b)) regardless of the images.

The mean squared error (MSE) was adopted to estimate the difference between the original and denoised values:

Joo et al.: Image Noise Reduction in Discrete Cosine Transform domain 22

1

2

0

1 ˆ(pN

i iip

)x yN

−

=

−∑ , (9)

where y is the estimated value, and is the number of pixels.

pN

Because the is insensitive to the noise level and image, 1.1 was adopted for

κκ . At this point, cycle

spinning for reducing the block artifact was applied, as shown in (10).

1 1

T, ,

0 0

1ˆ ( ( ( ( )))) ,N N

i j i ji j

x SN N λ

− −

− −= =

= Μ Μ× ∑∑ H H y (10)

where is the block size. N N× , ( )i jΜ

4. Experimental Results

0.8 1 1.2 1.4 1.680

100

120

140

160

180

200

κ

MS

Eκ vs MSE when σn = 20

BaboonBarbara

Boat

Cameraman

CapCarnival

Couple

EinsteinElaine

F16

GirlGoldhill

Lena

Mri

PentagonPeppers

(a)

0.8 1 1.2 1.4 1.635

40

45

50

55

60

65

κ

MS

E

κ vs MSE when σn = 10

BaboonBarbara

Boat

Cameraman

CapCarnival

Couple

EinsteinElaine

F16

GirlGoldhill

Lena

Mri

PentagonPeppers

(b)

Fig. 1. Optimal estimation (a) κ nσ =20, (b) nσ =10.

The effect of the block size on noise reduction performance was examined first. The block size of 4 × 4, 8 × 8, and 16 × 16 were tested through simulations, as shown in Table 1. The 8× 8 size is a trade-off between the computational complexity and energy compaction, and is also a reasonable size for noise separation, as shown in Table 1. Therefore, the 8× 8 block size was chosen for subsequent experiments. Four denoising methods were compared: 1) GGMRF-DCT method [3], 2) SADCT hard thresholding estimates [4], 3) BM3D [13], and 4) Bilateral filter [12], and the results are summarized in Table 2.

Table 1. Block size performance comparison.

PSNR (DB)

Block size Lena Pepper Barbara 4 32.30 31.50 30.29 8 33.42 32.22 31.72

16 33.15 31.84 31.89

Table 2. Noise reduction performance comparison.

PSNR (DB) 5nσ = Lena Pepper Barbara

Noisy 34.16 34.16 34.16 GGMRF-DCT 38.38 37.54 37.78

BM3D 38.72 37.63 38.31 SADCT 38.51 37.59 37.44

BF 37.29 37.12 36.00 Proposed 38.53 37.59 38.02

10nσ = Lena Pepper Barbara

Noisy 28.14 28.14 28.14 GGMRF-DCT 34.77 34.24 33.59

BM3D 35.93 35.01 34.98 SADCT 35.44 34.83 33.39

BF 33.53 33.71 31.36 Proposed 35.33 34.64 34.04

15nσ = Lena Pepper Barbara

Noisy 24.61 24.61 24.61 GGMRF-DCT 32.52 32.25 31.12

BM3D 34.27 33.72 33.11 SADCT 33.62 33.38 31.21

BF 31.35 31.67 28.78 Proposed 33.39 33.05 31.70

20nσ = Lena Pepper Barbara

Noisy 22.11 22.11 22.11 GGMRF-DCT 30.83 30.71 29.35

BM3D 33.05 32.76 31.78 SADCT 32.29 32.29 29.74

BF 29.79 30.08 27.09 Proposed 31.96 31.87 30.03

⋅ is a shift operator where i and j represent the horizontal and vertical shifts, respectively, which means cycle spinning in the horizontal and vertical direction. The inverse of the shift is

Hard thresholding is denoted as where for | |

1, ,( ) ( ).i j i j

−− −Μ ⋅ = Μ ⋅

( ) ,Sλ ⋅ ( )S w wλ = w λ> , and 0 otherwise. The denoising operation was applied to the N blocks represented as y In summary, DCT denoising by thres- holding, cycle spinning and averaging were applied to .

N×.

y

IEEK Transactions on Smart Processing and Computing, vol. 2, no. 1, February 2013 23

Table 3. Performance comparison for cycle spinning intervals.

PSNR (DB) CYCLE SPINNING INTERVAL D

image noisy 1 2 4 Lena 34.16 38.51 38.27 37.55

Pepper 34.16 36.91 36.71 36.02 5nσ =

Barbara 34.16 38.02 37.72 36.90 Lena 24.61 33.42 33.11 32.25

Pepper 24.61 32.06 31.94 31.14

For bilateral filtering, the window size was 11× 11 and the filtering parameters were and 2, dσ = 2r nσ σ= . The parameters, dσ and rσ , are the domain and range term parameters, respectively. The domain parameter is related to the spatial range of filtering, whereas the range term limits the difference in neighboring pixel intensity to be included in filtering. The test image size was 512× 512, and PSNR was used as a performance measure. The SADCT hard thresholding estimate performed well but required intensive calculations. This calculation requires adaptive LPA kernels for each pixel, followed by applying hard thresholding after shape adaptive DCT, whereas the proposed scheme was much simpler and achieved better performance. Furthermore, this method surpasses the GGMRF-DCT algorithm, which uses time consuming numerical iterations. The proposed method was slightly worse than BM3D, which is one of the best denoising methods currently available, even though they showed comparable performance when the noise level was low. BM3D is computationally expensive because it adopts collaborative filtering, which uses a 3-D transformation of a group by stacking similar image neighborhoods, thresholding of the transformation spectrum, and inverse 3-D transformation. In addition, this method develops the collaborative Wiener filtering. On the other hand, the proposed method is simple because it applies adaptive hard thresholding to the DCT coefficients while achieving high performance.

15nσ =

Barbara 24.61 31.72 31.31 30.33

After thresholding, one pixel is shifted for cycle spinning. The computation time can be reduced by increasing the shifting interval to two or four pixels. The interval d cycle spinning can be expressed as:

/ 1 / 1

T, ,

0 02

1ˆ ( ( ( ( ))))( / )

N d N d

d i d j d i d ji j

x SN d λ

− −

− × − × × ×= =

= Μ Μ∑ ∑ H H y ,

(11)

where denotes the cycle spinning interval. Table 3 and Fig. 2 show the effects of the cycle spinning shift intervals when the shifting interval is one, two or four pixels. Increasing the cycle spinning interval to two pixels reduces the computation time by a factor of four but the noise

reduction performance is similar. Therefore, a two pixel shift for cycle spinning is a reasonable choice.

d

Figs. 3 and 4 present the experimental results with magnified Pepper and Barbara, respectively. From the denoised images, the proposed method produced similar images to the originals. Although PSNR of the proposed method is slightly worse than BM3D, it is difficult to notice a difference in the subjective image quality.

5. Conclusion

A simple and effective denoising strategy based on the DCT thresholding technique was proposed. The presented approach is based on the noise energy for an adaptive hard threshold level of each DCT block. The low amplitude DCT coefficients are discarded until the sum of the thresholded signal is the same as the noise energy.

The proposed denoising algorithm is simple, yet highly effective compared to other published methods. In addition, although many other algorithms employ several parameters, the proposed method uses only the noise variance, which can be estimated reliably from the input image.

Integrating the proposed DCT-based denoising method with an existing platform would be an efficient and promising solution as DCT is used for JPEG and MPEG standards.

(a) (b) (c) (d)

Fig. 2. Performance comparison according to the cycle spinning intervals (a) noisy image when nσ =15, (b) denoised image when cycle spinning interval is 1, (c) when interval is 2, (d) when interval is 4.

Joo et al.: Image Noise Reduction in Discrete Cosine Transform domain 24

(a) (b) (c) (d)

(e) (f) (g)

Fig. 3. Experimental results (a) original cropped Pepper image, (b) noisy image when 20nσ = , denoised images by (c) GGMRF-DCT, (d) BM3D, (e) SADCT, (f) BL, (g) Proposed method.

(a) (b) (c) (d)

(e) (f) (g)

Fig. 4. Experimental results (a) original cropped Barbara image, (b) noisy image when 20nσ = , denoised images by (c) GGMRF-DCT, (d) BM3D, (e) SADCT, (f) BL, (g) Proposed method.

IEEK Transactions on Smart Processing and Computing, vol. 2, no. 1, February 2013 25

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Hyosun Joo received her B.S. and M.S. degree in Electronic Engineering from Ewha W. University in 2010 and 2012, respectively. She is currently working at SK Hynix Inc., where she is developing a CMOS image sensor. Her research interests include image

processing and computer vision.

Junhee Park received her B.S. and M.S. degree in Electronic Engineering from Ewha W. University in 2001 and 2003, respectively. She worked at the EtoM Solution and Korea Electric Test Institute as an engineer from 2003 to 2006. She received her Ph.D. degree in Electronic Engineering

from Ewha W. University in 2012. She is currently a post-doctoral researcher with the Image Processing Laboratory at Ewha W. University. Her research interests include image processing and computer vision.

Joo et al.: Image Noise Reduction in Discrete Cosine Transform domain 26

Jeongtae Kim received his B.S. and M.S. degree in Control and Instrumentation Engineering from Seoul National University, Seoul, Korea in 1989, 1991, respectively. From 1991 to 1998, he had worked for Samsung Electronics in Korea where he was engaged in the

development of the digital camcorder and digital TV. He received his Ph.D. degree in Electrical Engineering and Computer Science from the University of Michigan, Ann Arbor in 2004. Since 2004, he has been with the department of Electronics Engineering in Ewha W. University in Seoul, Korea, currently as an associate professor. His research interests include statistical signal processing, image restoration, image reconstruction, etc.

Byung-Uk Lee received his B.S. degree in Electronic Engineering from Seoul National University in 1979, and his M.S. degree in Electrical Science from Korea Advanced Institute of Science and Technology in 1981, and Ph.D. in Electrical Engineering from Stanford

University in 1991. He worked for Daewoo Electronics Co. from 1981 to 1985 and from 1991 to 1995. He joined the Department of Electronics Engineering, Ewha W. University in 1995, where he is currently a Professor. His research interests include image processing and computer vision.

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